This week, I want to talk about negative mass. Wait… negative mass? Like antimatter?

No.

Antimatter, in spite of its ‘anti’ name, still has a positive mass. An electron and an antielectron (positron) have opposite charge, but they have the same mass. Charge flips to negative, mass stays the same.

If antimatter is out of the picture — what in the world does have negative mass? Well, nothing really that we know of. It would take a very strange, exotic form of matter to have negative mass. The closest we come is with the complex Casimir Effect. But let’s imagine that a chunk of negative mass does exist. What do we know about it?

First off, negative masses would have… negative energies. When an object is moving, it has a kinetic energy equal to 1/2*m*v^2. If the mass is negative, the kinetic energy is negative. Even when you look at E = mc^2, the rest energy of matter, you’d find that the negative mass would lead to negative energy here, too!

Strange stuff. The faster it goes, the less energy it has. The more you have of it, the less energy you’ve got.

Gravity Time!

What would a negative mass mean for gravity? Well, we can check this out in the Gravity Simulator. Two planets with positive mass will, of course, attract.

There’s an extra bit of the puzzle we’ve left out. The force hasn’t changed, but the acceleration has. To figure out how something will move, we use F=ma. The force is equal to the mass times acceleration. Even if the force hasn’t changed in our example… if the sign of the mass has flipped, the sign of the acceleration has to flip. So instead of the pulling force making the planets accelerate together… they accelerate apart.

With all that in mind, the final puzzle is: what happens when you have a positive mass and a negative mass. Will they attract? Will they repel?

The test is easy to do (Note, you can do all these tests and MORE in the Gravity Simulator):

…whoa. Why did that happen? You’ve now got enough information to figure that one out.

As a final note, it would seem that having two planets suddenly zoom off the screen, ever faster, would violate Conservation of Momentum, or Conservation of Energy. But in fact, it doesn’t. As the positive mass gains positive kinetic energy, the negative mass gains negative kinetic energy. They cancel out.

Same with the momentum. Both planets may start moving in the same direction in that gif above, but the planet with negative mass has its momentum moving to the right. (Momentum = m*v, so flipping the sign of the mass flips the direction of the momentum). So even with strange negative masses around, conservation laws stay intact.

Will we ever really encounter a chunk of negative mass? Who can say. But at least we can make some good predictions about how we’d expect it to behave.

Let’s start off simple. The world we live in, for all practical purposes, is in three spatial dimensions. You can go up-down, left-right, or forward-backward. And in this world, Newton’s Law of Universal Gravitation tells us that two objects with mass will attract each other, according to:

If you only care about the distance, the force goes as 1/R^2, which is why we call this an inverse-squared law. Double the distance between the earth and the sun, and gravity will pull them together with 1/4 the strength. Inverse square laws are beautiful, because they lead to cool things like closed and stable orbits.

1/R^2 Force law. Ah, beautiful, closed orbits. Home.

All GREAT things for life!

Unfortunately, Newton’s Law of Gravity can’t help us when we’re in a world with a different number of dimensions. The law only works in 3D.

Gauss’s Law

Instead, our starting point instead is Gauss’s Law. Take a point mass, make an imaginary spherical surface around it. Imagine that instead of gravity, the point mass is just shooting out 100 bullets in random directions.

How many bullets pass through the sphere around the point mass? No matter how big or small you make your sphere, the answer will always be 100. That’s the core of Gauss’s Law. The number of bullets passing through a sphere doesn’t depend on its radius. When we’re talking about gravity, the bullets represent the Flux, which is simply the strength of gravity times the area of your sphere.

So then:

If the total flux is always the same, that means the Force of Gravity is proportional to 1/(Total Area of a Sphere). In 3D, a sphere has a surface area of 4PI*R^2. So the Force goes as the inverse of that, or 1/R^2.

2D

Suppose we were like the Flatlanders, and lived on a 2D plane. We could move forward-backward and left-right, but not up-down. How would that change gravity?

Well, Gauss’s Law still holds. But this time our ‘sphere’ that we draw around the point mass is actually just a circle. (Remember, we can’t leave our 2D surface!) The ‘surface area’ of the circle is just its circumference: 2PI*R. Which means that the Force of Gravity goes as 1/R. And we get orbits that look like this:

1/R Force Law. Notice that the orbits don’t match up nicely anymore.

4D

What if there were an extra dimension? Suppose we lived in a world where we could move up-down, left-right, forward-backward, and… and… uh… 4Up-4Down. We don’t have a word for those last two of course, since we never move through a fourth spatial dimension. But nonetheless, we are armed with all the information we need to figure out how gravity would look.

This time the ‘sphere’ we make around a point charge is in 4D, which is hard to draw, but we can figure out what it must be like with a simple pattern. In 2D, the ‘surface’ of a sphere is a line. In 3D, the ‘surface’ of a sphere is an area. So in 4D the ‘surface’ of a sphere should be a volume. Its units should be length cubed. No surprise, then, that the surface ‘area’ of the 4D sphere is 2*PI^2*R^3. Which means that the Force of Gravity must go as 1/R^3. So we get orbits like this:

A 1/r^3 Force Law. Yikes, I’d hate to be on *any* of those planets.

Beyond

You’ve now got everything you need to ponder the force of gravity in any number of dimensions. And with the Gravity Simulator, you can check out what any of these worlds might look like. Try to make a 5D solar system, I challenge you!

This was just a quick introduction to some interesting topics — so if you wanna know more, there are plenty of resources out there you can use.